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A chain rule formula for the composition of a vector-valued function by a piecewise smooth function

François Murat, Cristina Trombetti (2003)

Bollettino dell'Unione Matematica Italiana

We state and prove a chain rule formula for the composition T u of a vector-valued function u W 1 , r Ω ; R M by a globally Lipschitz-continuous, piecewise C 1 function T . We also prove that the map u T u is continuous from W 1 , r Ω ; R M into W 1 , r Ω for the strong topologies of these spaces.

A criterion of Γ-nullness and differentiability of convex and quasiconvex functions

Jaroslav Tišer, Luděk Zajíček (2015)

Studia Mathematica

We introduce a criterion for a set to be Γ-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Fréchet differentiable is Γ-null. Our criterion also implies a new result about Gâteaux (and Hadamard) differentiability of quasiconvex functions.

A Lipschitz function which is C on a.e. line need not be generically differentiable

Luděk Zajíček (2013)

Colloquium Mathematicae

We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is C smooth on a.e. line parallel to v and f is Gâteaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dimX > 1) is an arbitrary Banach space and “a.e.” has any usual “measure sense”. This example gives an answer to a natural question concerning the author’s recent study of linearly essentially smooth functions (which generalize essentially smooth...

A new proof of Fréchet differentiability of Lipschitz functions

Joram Lindenstrauss, David Preiss (2000)

Journal of the European Mathematical Society

We give a relatively simple (self-contained) proof that every real-valued Lipschitz function on 2 (or more generally on an Asplund space) has points of Fréchet differentiability. Somewhat more generally, we show that a real-valued Lipschitz function on a separable Banach space has points of Fréchet differentiability provided that the w * closure of the set of its points of Gâteaux differentiability is norm separable.

A note on a class of homeomorphisms between Banach spaces

Piotr Fijałkowski (2005)

Colloquium Mathematicae

This paper deals with homeomorphisms F: X → Y, between Banach spaces X and Y, which are of the form F ( x ) : = F ̃ x ( 2 n + 1 ) where F ̃ : X 2 n + 1 Y is a continuous (2n+1)-linear operator.

A Note on Differentiability of Lipschitz Maps

Rafał Górak (2010)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that every Lipschitz map defined on an open subset of the Banach space C(K), where K is a scattered compactum, with values in a Banach space with the Radon-Nikodym property, has a point of Fréchet differentiability. This is a strengthening of the result of Lindenstrauss and Preiss who proved that for countable compacta. As a consequence of the above and a result of Arvanitakis we prove that Lipschitz functions on certain function spaces are Gâteaux differentiable.

A note on intermediate differentiability of Lipschitz functions

Luděk Zajíček (1999)

Commentationes Mathematicae Universitatis Carolinae

Let f be a Lipschitz function on a superreflexive Banach space X . We prove that then the set of points of X at which f has no intermediate derivative is not only a first category set (which was proved by M. Fabian and D. Preiss for much more general spaces X ), but it is even σ -porous in a rather strong sense. In fact, we prove the result even for a stronger notion of uniform intermediate derivative which was defined by J.R. Giles and S. Sciffer.

A one-sided version of Alexiewicz-Orlicz's differentiability theorem.

E. Corbacho, A. Plichko, V. Tarieladze (2005)


Modificando adecuadamente el método de un trabajo olvidado [1], probamos que si una aplicación continua, de un subconjunto abierto no vacío U de un espacio vectorial topológico metrizable separable y de Baire E, en un espacio localmente convexo, es direccionalmente diferenciable por la derecha en U según un subconjunto comagro de E, entonces, es genéricamente Gâteaux diferenciable en U. Nuestro resultado implica que cualquier espacio vectorial topológico, metrizable, separable y de Baire, es débilmente...

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